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arXiv · 2510.03927

Analysis of Compact Symmetric Finite Difference Methods with Highest Possible Orders for Elliptic Equations

Qiwei Feng·Bin Han·Michelle Michelle·Jiwoon Sim
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Abstract

For the $d$-dimensional variable-coefficient Poisson problem $-\nabla \cdot(a\nabla u)=f$ subject to Dirichlet boundary conditions, we seek compact (i.e., $3^d$-point) symmetric finite difference methods (FDMs) of the highest possible consistency orders, which yield symmetric positive definite (SPD) linear systems so that many fast solvers (e.g., the conjugate gradient method) can be used. Although 2nd-order compact symmetric FDMs for variable-coefficient Poisson equations are known, the construction of higher-order counterparts has, to the best of our knowledge, remained an open problem. This is largely because all stencils must satisfy highly specific interdependent relations and high-order consistency imposes numerous local technical conditions. This paper provides a comprehensive characterization of the highest consistency order that a $d$-dimensional compact symmetric FDM can achieve for variable-coefficient Poisson equations on a uniform grid. For the 1D case, we prove that our symmetric compact FDMs on a uniform grid can achieve arbitrary consistency orders, yield tridiagonal SPD matrices, and converge at the corresponding orders in the $\infty$-norm and the weighted $2$-norm. For any $d$-dimensional case with $d\ge 2$, we prove that compact (not necessarily symmetric) FDMs can achieve 6th-order consistency and that no higher consistency order is possible. If $\tfrac{|\nabla a|^2}{a^2}-\tfrac{2Δa}{a}$ is nonconstant, $d$-dimensional compact symmetric FDMs can achieve 4th-order consistency, but no higher. On the other hand, if $\tfrac{|\nabla a|^2}{a^2}-\tfrac{2Δa}{a}$ is constant, then $d$-dimensional compact symmetric FDMs can achieve 6th-order consistency, which is the highest possible order. For $d=2,3,4$ and $M=4,6$, we prove that our proposed FDMs produce SPD matrices, and achieve $M$th-order convergence in the weighted 2-norm and the $\infty$-norm.

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Qiwei Feng, Bin Han, Michelle Michelle, Jiwoon Sim. 2026-09-16. Analysis of Compact Symmetric Finite Difference Methods with Highest Possible Orders for Elliptic Equations. https://arxiv.org/abs/2510.03927

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